Colloidal nanocrystals (NCs) of APbX₃-type lead halide perovskites [A = Cs⁺, CH₃NH₃⁺ (methylammonium or MA⁺) or CH(NH₂)₂⁺ (formamidinium or FA⁺); X = Cl^-, Br^-, I^-] have recently emerged as highly versatile photonic sources for applications ranging from simple photoluminescence down-conversion (e.g., for display backlighting) to light-emitting diodes. From the perspective of spectral coverage, a formidable challenge facing the use of these materials is how to obtain stable emissions in the red and infrared spectral regions covered by the iodide-based compositions. So far, red-emissive CsPbI₃ NCs have been shown to suffer from a delayed phase transformation into a nonluminescent, wide-band-gap 1D polymorph, and MAPbI₃ exhibits very limited chemical durability. In this work, we report a facile colloidal synthesis method for obtaining FAPbI₃ and FA-doped CsPbI₃ NCs that are uniform in size (10-15 nm) and nearly cubic in shape and exhibit drastically higher robustness than their MA- or Cs-only cousins with similar sizes and morphologies. Detailed structural analysis indicated that the FAPbI₃ NCs had a cubic crystal structure, while the FA_0.1Cs_0.9PbI₃ NCs had a 3D orthorhombic structure that was isostructural to the structure of CsPbBr₃ NCs. Bright photoluminescence (PL) with high quantum yield (QY > 70%) spanning red (690 nm, FA_0.1Cs_0.9PbI₃ NCs) and near-infrared (near-IR, ca. 780 nm, FAPbI₃ NCs) regions was sustained for several months or more in both the colloidal state and in films. The peak PL wavelengths can be fine-tuned by using postsynthetic cation- and anion-exchange reactions. Amplified spontaneous emissions with low thresholds of 28 and 7.5 μJ cm^-2 were obtained from the films deposited from FA_0.1Cs_0.9PbI₃ and FAPbI₃ NCs, respectively. Furthermore, light-emitting diodes with a high external quantum efficiency of 2.3% were obtained by using FAPbI₃ NCs.